## Abstract

We study the symplectic mapping class groups of (CP^{2}#5CP^{2}, ω). Our main innovation is to avoid the detailed analysis of the topology of generic almost complex structures J_{0} as in most of earlier literature. Instead, we use a combination of the technique of ball-swapping (defined by Wu [Math. Ann. 359 (2014), pp. 153–168]) and the study of a semi-toric model to understand a “connecting map”, whose cokernel is the symplectic mapping class group. Using this approach, we completely determine the Torelli symplectic mapping class group (Torelli SMCG) for all symplectic forms ω. Let N_{ω} be the number of (−2)-symplectic spherical homology classes. Torelli SMCG is trivial if N_{ω} > 8; it is π0(Diff^{+}(S^{2}, 5)) if N_{ω} = 0 (by Seidel [Lecture notes in Math., Springer, Berlin, 2008] and Evans [J. Symplectic Geom. 9 (2011), pp. 45–82]); and it is π0(Diff^{+}(S^{2}, 4)) in the remaining case. Further, we completely determine the rank of π1(Symp(CP^{2}#5CP^{2}, ω)) for any given symplectic form. Our results can be uniformly presented in terms of Dynkin diagrams of type A and type D Lie algebras. We also provide a solution to the smooth isotopy problem of rational 4-manifolds (open problem 16 in McDuff-Salamon’s book 3rd version [Oxford graduate texts in mathematics, Oxford University Press, Oxford, 2017]).

Original language | English (US) |
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Pages (from-to) | 1357-1410 |

Number of pages | 54 |

Journal | Transactions of the American Mathematical Society |

Volume | 375 |

Issue number | 2 |

DOIs | |

State | Published - 2022 |

### Bibliographical note

Funding Information:Received by the editors February 10, 2020, and, in revised form, June 12, 2021, and July 8, 2021. 2020 Mathematics Subject Classification. Primary 57R17, 53D35; Secondary 14D22, 57S05. The first author was supported by NSF Grants and an AMS-Simons travel grant. The second author was supported by NSF Grants. The third author was partially supported by Simons Collaboration Grant 524427.

Publisher Copyright:

© 2021 American Mathematical Society.